Introduction

The human eye can perceive a narrow range of electromagnetic radiation, which is known as light. This visible light spectrum typically spans wavelengths from approximately 400 nm to 750 nm.

Based on everyday observations, we understand two key properties of light: its incredibly high speed and its tendency to travel in straight lines. While the straight-line travel is an intuitive concept, the finite nature and immense speed of light were discoveries made over time. The accepted speed of light in a vacuum is $c \approx 3 \times 10^8 \text{ m s}^{-1}$, representing the maximum speed attainable in nature.

The concept of light traveling in straight lines appears to contrast with its nature as an electromagnetic wave. This apparent contradiction is resolved by considering the relative sizes of light wavelengths and the objects it interacts with. When light's wavelength is significantly smaller than the dimensions of objects it encounters (which is true for visible light and everyday objects), the wave propagation can be effectively approximated as travel along a straight path. This straight path is termed a ray of light, and a collection of such rays forms a beam of light.

This chapter explores various optical phenomena, including reflection, refraction, and dispersion, by employing this ray picture of light. We will apply the fundamental laws of reflection and refraction to understand how images are formed by both flat and curved mirrors and lenses. Furthermore, we will investigate the construction and operation of common optical devices, including the human eye, microscopes, and telescopes.

Reflection Of Light By Spherical Mirrors

The behaviour of light when it bounces off a surface is governed by the laws of reflection. These laws state that:

1. The angle of incidence (the angle between the incoming ray and the perpendicular line to the surface, called the normal) is always equal to the angle of reflection (the angle between the outgoing ray and the normal).

2. The incident ray, the reflected ray, and the normal to the reflecting surface at the point where the light hits, all lie within the same plane.

These fundamental laws are applicable at every point on any reflecting surface, regardless of whether it is flat or curved. Our focus here will be on spherical mirrors, which are parts of a hollow sphere. For a spherical surface, the normal at any point is the line joining that point to the center of the sphere from which the mirror is cut. This line is equivalent to the radius at that point.

Key terms associated with spherical mirrors include:

• Pole (P): The geometrical center of the spherical mirror's reflecting surface.
• Centre of Curvature (C): The center of the imaginary sphere of which the mirror is a part.
• Principal Axis: The straight line passing through the pole and the center of curvature of the spherical mirror.

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Sign Convention

To consistently apply mathematical formulas for mirrors and lenses, a standard system for measuring distances and heights is essential. We follow the Cartesian sign convention:

1. All distances are measured from the pole (P) of the mirror or the optical center of a lens.

2. Distances measured in the same direction as the incoming (incident) light are considered positive.

3. Distances measured in the direction opposite to the incoming (incident) light are considered negative.

4. Heights measured upwards from and perpendicular to the principal axis are considered positive.

5. Heights measured downwards from and perpendicular to the principal axis are considered negative.

Adopting this convention ensures that a single formula can describe image formation for various cases, including different types of mirrors/lenses and real/virtual images.

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Focal Length Of Spherical Mirrors

When a beam of light rays traveling parallel to the principal axis strikes a spherical mirror, these rays either converge to a point or appear to diverge from a point after reflection. This point is called the principal focus (F).

• For a concave mirror, parallel paraxial rays converge to a real focus point on the principal axis in front of the mirror.
• For a convex mirror, parallel paraxial rays diverge such that they appear to originate from a virtual focus point located behind the mirror on the principal axis.

The focal plane is a plane passing through the principal focus and perpendicular to the principal axis. If parallel rays are incident at an angle to the principal axis, they converge to a point (or appear to diverge from a point) in the focal plane.

The distance between the pole (P) and the principal focus (F) is called the focal length (f) of the mirror. For spherical mirrors with a small aperture compared to their radius of curvature (paraxial approximation), the focal length is related to the radius of curvature ($R$) by the formula:

$$f = \frac{R}{2}$$

This relationship can be derived using geometric principles and the laws of reflection, assuming that the incident rays are paraxial (close to the principal axis and making small angles). In the paraxial approximation, trigonometric functions of small angles can be approximated by the angle itself ($\tan \theta \approx \theta$), and distances measured along the principal axis can be approximated by distances from the pole.

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The Mirror Equation

An image is formed at a point where light rays emanating from an object point actually meet after reflection or refraction (real image) or appear to meet after being extended backwards (virtual image). Image formation establishes a correspondence between points on the object and points on the image.

To find the image of an object formed by a spherical mirror, we can trace the path of specific rays originating from the object. Convenient rays to trace include:

1. A ray parallel to the principal axis reflects through the principal focus (concave) or appears to come from the principal focus (convex).

2. A ray passing through the center of curvature (concave) or directed towards the center of curvature (convex) strikes the mirror normally and reflects back along its original path.

3. A ray passing through the principal focus (concave) or directed towards the principal focus (convex) reflects parallel to the principal axis.

4. A ray incident at the pole reflects according to the law of reflection, with the principal axis acting as the normal.

Tracing at least two of these rays from a point on the object helps locate the corresponding image point. The image of an extended object can be found by locating the images of several points on the object (e.g., the endpoints).

The mathematical relationship between the object distance ($u$), image distance ($v$), and focal length ($f$) for a spherical mirror is given by the mirror equation:

$$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$$

Here, $u$ is the distance of the object from the pole, and $v$ is the distance of the image from the pole. Distances must be used with appropriate signs based on the Cartesian sign convention.

The linear magnification (m) quantifies how much the image is enlarged or reduced compared to the object. It is defined as the ratio of the height of the image ($h'$) to the height of the object ($h$):

$$m = \frac{h'}{h}$$

Using similar triangles formed by the object, image, and principal axis, the magnification can also be related to the image and object distances:

$$m = -\frac{v}{u}$$

The sign of $m$ indicates the image orientation: a positive $m$ signifies an erect image, while a negative $m$ indicates an inverted image. These equations ($1/v + 1/u = 1/f$ and $m = -v/u$) are valid for all types of spherical mirrors (concave and convex) and for both real and virtual images when the sign convention is applied correctly.

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Refraction

When a beam of light transitions from one transparent medium into another, it undergoes a change in direction at the boundary separating the two media. This phenomenon is called refraction. A portion of the light is also reflected back into the original medium (this is called internal reflection when light goes from denser to rarer), but the bending of light as it enters the new medium is the defining characteristic of refraction.

For light incident obliquely (at an angle other than 0° or 90°), the direction of propagation changes. The fundamental rules governing refraction are known as Snell's laws of refraction, based on experimental observations:

1. The incident ray, the refracted ray, and the normal (the line perpendicular to the interface) at the point where the light hits, all lie in the same plane.

2. For a given pair of media and a specific colour (wavelength) of light, the ratio of the sine of the angle of incidence ($\sin i$) to the sine of the angle of refraction ($\sin r$) is a constant. This constant is called the refractive index of the second medium with respect to the first medium, denoted as $n_{21}$.

$$\frac{\sin i}{\sin r} = n_{21}$$

If $n_{21} > 1$, then $\sin i > \sin r$. Since the sine function increases with angle in the relevant range (0° to 90°), this implies $i > r$. In this case, the refracted ray bends towards the normal. Medium 2 is considered optically denser than medium 1.

If $n_{21} < 1$, then $\sin i < \sin r$, implying $i < r$. The refracted ray bends away from the normal. Medium 2 is considered optically rarer than medium 1.

It's important to distinguish between optical density and mass density. Optical density is related to how much a medium slows down light (inversely proportional to the speed of light in the medium), while mass density is mass per unit volume. A medium with higher mass density is not necessarily optically denser. For example, turpentine has lower mass density than water but is optically denser, causing light to bend more towards the normal when entering it from air compared to water.

The refractive index of medium 1 with respect to medium 2, $n_{12}$, is the reciprocal of the refractive index of medium 2 with respect to medium 1, $n_{21}$:

$$n_{12} = \frac{1}{n_{21}}$$

Refractive indices are relative, but often we refer to the absolute refractive index of a medium, which is its refractive index with respect to vacuum or air (which has a refractive index very close to 1). Let $n_1$ be the absolute refractive index of medium 1 and $n_2$ be the absolute refractive index of medium 2. Then $n_{21} = n_2/n_1$. Snell's law can be written as $n_1 \sin i = n_2 \sin r$. For a chain of media 1, 2, 3, the relationship between refractive indices is $n_{31} = n_{32} \times n_{21}$, where $n_{31} = n_3/n_1$, $n_{32} = n_3/n_2$, and $n_{21} = n_2/n_1$. This shows $(n_3/n_1) = (n_3/n_2) \times (n_2/n_1)$, which is consistent.

Simple consequences of refraction include the lateral shift of light passing through a parallel-sided glass slab (the emergent ray is parallel to the incident ray but displaced sideways) and the phenomenon of objects appearing shallower when viewed through a denser medium like water. For viewing nearly perpendicular to the surface, the apparent depth ($h_{apparent}$) is related to the real depth ($h_{real}$) and the refractive index ($n$) of the medium by:

$$h_{apparent} = \frac{h_{real}}{n}$$

Total Internal Reflection

When light travels from an optically denser medium to an optically rarer medium (e.g., from water to air), it undergoes refraction and bends away from the normal. Simultaneously, a portion of the light is reflected back into the denser medium; this is called internal reflection.

As the angle of incidence ($i$) in the denser medium increases, the angle of refraction ($r$) in the rarer medium also increases (since $r > i$). There comes a specific angle of incidence for which the refracted ray grazes the boundary between the two media, meaning the angle of refraction is 90°. This particular angle of incidence is known as the critical angle ($i_c$).

According to Snell's law, at the critical angle: $n_{denser} \sin i_c = n_{rarer} \sin 90^\circ$. Since $\sin 90^\circ = 1$, we get:

$$\sin i_c = \frac{n_{rarer}}{n_{denser}} = \frac{1}{n_{denser, rarer}}$$

where $n_{denser, rarer}$ is the refractive index of the denser medium with respect to the rarer medium ($n_{denser}/n_{rarer}$).

If the angle of incidence ($i$) in the denser medium becomes greater than the critical angle ($i > i_c$), refraction is no longer possible. All the light is reflected back into the denser medium. This phenomenon is called Total Internal Reflection (TIR). Unlike normal reflection where some light is always transmitted, in TIR, there is theoretically no loss of light due to transmission, resulting in a highly intense reflected ray.

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Total Internal Reflection In Nature And Its Technelogical Applications

Total Internal Reflection is responsible for several fascinating natural phenomena and is extensively used in modern technology:

1. Mirage: An atmospheric optical phenomenon where light is refracted to produce a displaced image of distant objects or the sky. It occurs when light passes through layers of air with different temperatures and thus different refractive indices, leading to TIR, especially in hot conditions near the ground.

2. Brilliance of Diamonds: Diamonds sparkle brightly due to their very high refractive index (around 2.42) which results in a small critical angle (about 24.4°). When light enters a diamond, it undergoes multiple total internal reflections inside before emerging, giving it its characteristic brilliance.

3. Prisms: Right-angled prisms are designed with angles such that light entering one face undergoes TIR at another face, allowing them to deviate light by 90° or 180° without significant loss of intensity. They can also be used to invert images without changing their size. For this, the critical angle of the prism material must be less than 45°.

4. Optical Fibres: These are thin strands of high-quality glass or quartz that transmit light signals over long distances using TIR. An optical fibre consists of a core (higher refractive index) surrounded by cladding (lower refractive index). Light entering the core at a suitable angle strikes the core-cladding interface at an angle greater than the critical angle and undergoes repeated TIR along the length of the fibre, even when the fibre is bent. This mechanism ensures minimal signal loss. Optical fibres are crucial for high-speed internet, telecommunications, and are also used in medicine (endoscopes) and decorative lighting.

Here is a table showing critical angles for some substances with respect to air:

Substance medium	Refractive index	Critical angle
Water	1.33	48.75°
Crown glass	1.52	41.14°
Dense flint glass	1.62	37.31°
Diamond	2.42	24.41°

Refraction At Spherical Surfaces And By Lenses

Extending the principles of refraction from plane surfaces, we can analyze how light bends when it passes through a curved boundary between two transparent media. A small section of a spherical surface can be treated as planar, and Snell's laws apply at each point of incidence. For a spherical surface, the normal at any point is a line passing through the center of curvature of the sphere.

Thin lenses are optical components made of transparent material (like glass or plastic) bounded by two surfaces, at least one of which is spherical. Common types are biconvex, biconcave, plano-convex, plano-concave, etc. Image formation by a lens can be understood as the result of two successive refractions, one at each spherical surface.

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Refraction At A Spherical Surface

Consider a spherical surface separating two media with refractive indices $n_1$ and $n_2$. Let the center of curvature be C and the radius of curvature be R. For paraxial rays originating from an object point O in medium 1 and forming an image point I in medium 2, the relationship between object distance ($u$, measured from the surface), image distance ($v$, measured from the surface), refractive indices ($n_1$, $n_2$), and radius of curvature ($R$) is given by the formula:

$$\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}$$

This formula holds for any spherical refracting surface (convex or concave) and applies with proper use of the Cartesian sign convention for distances $u$, $v$, and $R$. $u$ is typically negative for a real object in front of the surface (incident light direction is positive), $v$ is positive for a real image formed on the side of refracted light, and $R$ is positive for a convex surface (center of curvature on the side of refracted light) and negative for a concave surface (center of curvature on the side of incident light), following the standard convention relative to the pole.

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Refraction By A Lens

Image formation by a thin lens (a lens where the thickness is negligible compared to object/image distances and radii of curvature) can be analyzed by considering refraction at each of its two surfaces sequentially. Light from an object first refracts at the front surface, forming an intermediate image. This intermediate image then acts as the object (either real or virtual) for the second surface, which performs another refraction to form the final image.

Applying the spherical surface refraction formula $\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}$ to each surface of a thin lens and combining the results leads to the lens maker's formula:

$$\frac{1}{f} = \left(\frac{n_{lens}}{n_{medium}} - 1\right) \left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$

Here, $f$ is the focal length of the lens, $n_{lens}$ is the refractive index of the lens material, $n_{medium}$ is the refractive index of the surrounding medium, $R_1$ is the radius of curvature of the first surface the light encounters, and $R_2$ is the radius of curvature of the second surface. $R_1$ and $R_2$ are used with their proper signs according to the convention ($R$ is positive if the center of curvature is on the side of light travel after refraction from that surface, and negative otherwise).

The focal length $f$ is a property of the lens and the medium it's in. It's positive for a converging lens (like a convex lens in air) and negative for a diverging lens (like a concave lens in air). The lens maker's formula is useful for designing lenses with desired focal lengths.

For a thin lens, the relationship between the object distance ($u$), image distance ($v$), and focal length ($f$) is given by the thin lens formula:

$$\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$$

This formula is universally applicable to all thin lenses (convex and concave) and all types of images (real and virtual) when distances are substituted with appropriate signs. The object distance $u$ is measured from the optical center (the central point of the thin lens), and the image distance $v$ is also measured from the optical center.

Similar to mirrors, specific rays can be traced to determine the image location and characteristics for lenses:

1. A ray parallel to the principal axis passes through the second principal focus F' after refraction (convex lens) or appears to diverge from the first principal focus F (concave lens).

2. A ray passing through the optical center (the geometric center of the thin lens) travels undeviated.

3. A ray passing through the first principal focus F (convex lens) or directed towards the second principal focus F' (concave lens) emerges parallel to the principal axis after refraction.

The magnification (m) produced by a lens is defined as the ratio of image height ($h'$) to object height ($h$). For a thin lens, this is related to image and object distances by:

$$m = \frac{h'}{h} = \frac{v}{u}$$

The sign of $m$ indicates the image orientation: positive for erect, negative for inverted.

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Power Of A Lens

The ability of a lens to converge or diverge light rays incident upon it is quantified by its power (P). A lens with a shorter focal length bends light more strongly and thus has greater power.

Power is defined as the reciprocal of the focal length, provided the focal length is measured in meters:

$$P = \frac{1}{f}$$

The standard unit for lens power is the dioptre (D). One dioptre is the power of a lens with a focal length of 1 meter ($1 D = 1 \text{ m}^{-1}$).

Converging lenses (like convex lenses in air) have a positive focal length and therefore positive power. Diverging lenses (like concave lenses in air) have a negative focal length and thus negative power.

For instance, a prescription of +2.5 D indicates a convex lens with a focal length of $1/2.5 \text{ m} = 0.4 \text{ m} = 40 \text{ cm}$. A prescription of -4.0 D indicates a concave lens with a focal length of $1/(-4.0) \text{ m} = -0.25 \text{ m} = -25 \text{ cm}$.

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Combination Of Thin Lenses In Contact

When two or more thin lenses are placed in contact with each other such that their principal axes coincide, they form a lens system. The effect of this combination can often be represented by a single equivalent lens with a specific focal length and power.

Consider two thin lenses A and B with focal lengths $f_1$ and $f_2$ in contact. An object O is placed in front of the first lens A. Lens A forms an image I$_1$. If lens B were not present, this image would be formed at a distance $v_1$ from the lenses. Using the thin lens formula for lens A: $\frac{1}{v_1} - \frac{1}{u} = \frac{1}{f_1}$.

This intermediate image I$_1$ now serves as the object for the second lens B. The object distance for lens B is $u_2 = v_1$. If the intermediate image is formed beyond lens B, it is a virtual object for lens B (positive object distance in this convention). If it's formed before or at lens B, it's a real object (negative object distance). The second lens B forms the final image I at a distance $v$ from the lenses. Using the thin lens formula for lens B: $\frac{1}{v} - \frac{1}{v_1} = \frac{1}{f_2}$.

Adding the equations for lens A and lens B:

$$\left(\frac{1}{v_1} - \frac{1}{u}\right) + \left(\frac{1}{v} - \frac{1}{v_1}\right) = \frac{1}{f_1} + \frac{1}{f_2}$$

$$\frac{1}{v} - \frac{1}{u} = \frac{1}{f_1} + \frac{1}{f_2}$$

If we consider the combination as a single equivalent thin lens with focal length $f_{eq}$, its lens formula is $\frac{1}{v} - \frac{1}{u} = \frac{1}{f_{eq}}$. Comparing the equations, we get the formula for the effective focal length of two thin lenses in contact:

$$\frac{1}{f_{eq}} = \frac{1}{f_1} + \frac{1}{f_2}$$

For a combination of several thin lenses in contact with focal lengths $f_1, f_2, f_3, \dots$, the effective focal length is given by:

$$\frac{1}{f_{eq}} = \frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3} + \dots$$

In terms of power, since $P = 1/f$, the total power of the combination is the algebraic sum of the individual lens powers:

$$P_{total} = P_1 + P_2 + P_3 + \dots$$

This addition is algebraic, meaning powers of convex lenses are positive and powers of concave lenses are negative. Combining lenses allows for achieving desired magnification, image orientation, and correcting optical defects (aberrations).

The total magnification produced by a combination of lenses is the product of the magnifications produced by each lens individually:

$$m_{total} = m_1 \times m_2 \times m_3 \times \dots$$

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Refraction Through A Prism

A prism is a transparent optical element with flat, polished surfaces that refract light. Typically, it has a triangular cross-section with two refracting surfaces inclined at an angle to each other, called the angle of the prism (A). When a ray of light passes through a prism, it undergoes two refractions, one at each face.

Consider a ray incident on one face at an angle $i$. It refracts into the prism at an angle $r_1$. Inside the prism, it travels to the second face and is incident upon it at an angle $r_2$. It then refracts out of the prism at an angle $e$ (angle of emergence).

The total change in the direction of the ray as it passes through the prism is called the angle of deviation ($\delta$). It is the angle between the emergent ray and the direction of the incident ray. The angles involved are related by the following geometric principles:

1. The angle of the prism ($A$) is related to the angles of refraction inside the prism: $A = r_1 + r_2$.

2. The total deviation ($\delta$) is the sum of the deviations at each surface: $\delta = (i - r_1) + (e - r_2) = i + e - (r_1 + r_2)$. Substituting the first relation, we get $\delta = i + e - A$.

The angle of deviation depends on the angle of incidence, the angle of the prism, and the refractive index of the prism material. As the angle of incidence varies, the deviation angle changes, reaching a minimum value called the angle of minimum deviation ($D_m$).

At minimum deviation, the following conditions hold:

1. The angle of incidence equals the angle of emergence ($i = e$).

2. The angles of refraction inside the prism are equal ($r_1 = r_2$).

From $A = r_1 + r_2$ and $r_1 = r_2$, we get $A = 2r$, so $r = A/2$.

From $\delta = i + e - A$ and $\delta = D_m$, $i = e$, we get $D_m = 2i - A$, so $i = (A + D_m)/2$.

Applying Snell's law ($n_1 \sin i = n_2 \sin r$) at the first face (assuming medium 1 is air with $n_1 \approx 1$ and medium 2 is prism material with refractive index $n$):

$$1 \times \sin i = n \sin r_1$$

Substituting the expressions for $i$ and $r$ at minimum deviation:

$$n = \frac{\sin\left(\frac{A + D_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$$

This formula is used to determine the refractive index of the prism material by measuring the angle of the prism ($A$) and the angle of minimum deviation ($D_m$).

For a thin prism (a prism with a very small angle $A$), the angle of minimum deviation $D_m$ is also small. Using the small angle approximation for sine ($\sin \theta \approx \theta$), the formula simplifies to:

$$n \approx \frac{(A + D_m)/2}{A/2} = \frac{A + D_m}{A} = 1 + \frac{D_m}{A}$$

Rearranging, the deviation for a thin prism is approximately:

$$\delta \approx (n - 1)A$$

Thin prisms produce small deviations and do not cause significant dispersion (splitting of white light into colours) compared to thicker prisms.

Optical Instruments

Various optical devices and instruments have been developed to extend or improve the capabilities of human vision, utilizing the principles of reflection and refraction. These include simple tools like periscopes and kaleidoscopes, as well as complex instruments like binoculars, telescopes, and microscopes. While the human eye itself is a sophisticated natural optical instrument, we will focus on the working principles of the microscope and the telescope here.

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The Microscope

Microscopes are used to obtain magnified images of small objects. A basic type is the simple microscope, which is essentially a single converging lens (a convex lens) with a short focal length. To use it as a magnifier, the object is placed within or very close to the focal point of the lens. The lens forms a virtual, erect, and magnified image.

The image can be formed either at the near point (D), which is the least distance of distinct vision (approximately 25 cm for a normal eye), or at infinity (for relaxed viewing).

The magnification (m) for a simple microscope is typically referred to as angular magnification or magnifying power. It's the ratio of the angle subtended by the image at the eye to the angle subtended by the object if it were placed at the near point D and viewed directly.

When the final image is formed at the near point (D), the magnifying power is:

$$m = 1 + \frac{D}{f}$$

When the final image is formed at infinity (relaxed eye viewing), the magnifying power is:

$$m = \frac{D}{f}$$

Here, $f$ is the focal length of the lens and $D \approx 25 \text{ cm}$. A simple microscope provides a limited magnification, typically up to about 9x.

For achieving much higher magnifications, a compound microscope is used. It consists of two converging lenses:

1. The Objective Lens: Placed close to the object, it has a very short focal length ($f_o$). It forms a real, inverted, and magnified intermediate image.

2. The Eyepiece (or Ocular): Acts like a simple microscope, magnifying the intermediate image formed by the objective. It also has a short focal length ($f_e$). It forms the final, enlarged, virtual image. This final image is inverted with respect to the original object.

The intermediate image is usually positioned near or within the focal plane of the eyepiece. The distance between the second focal point of the objective and the first focal point of the eyepiece is called the tube length (L).

The total magnifying power ($m$) of a compound microscope is the product of the linear magnification produced by the objective ($m_o$) and the angular magnification produced by the eyepiece ($m_e$):

$$m = m_o \times m_e$$

The linear magnification of the objective is approximately $m_o \approx \frac{\text{distance of intermediate image from objective}}{\text{object distance from objective}}$. For the intermediate image formed near the eyepiece's focus, this is approximately $m_o \approx L/f_o$.

The angular magnification of the eyepiece is like that of a simple microscope:

• If the final image is at the near point: $m_e = 1 + D/f_e$
• If the final image is at infinity: $m_e = D/f_e$

Thus, for the final image at infinity, the total magnifying power is approximately:

$$m \approx \left(\frac{L}{f_o}\right) \left(\frac{D}{f_e}\right)$$

To achieve high magnification, both the objective ($f_o$) and the eyepiece ($f_e$) should have small focal lengths. Modern microscopes use multiple lens elements in both the objective and eyepiece to minimize optical aberrations and improve image quality.

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Telescope

Telescopes are used to view distant objects, providing angular magnification so that these objects appear closer and larger. Similar to a compound microscope, a telescope also has two primary lenses:

1. The Objective Lens: This lens is placed towards the distant object and has a large focal length ($f_o$) and a large aperture (diameter). It collects light from the distant object and forms a real, inverted, and diminished intermediate image at its focal point.

2. The Eyepiece: This lens is placed near the observer's eye and has a short focal length ($f_e$). It acts as a magnifier, magnifying the intermediate image formed by the objective. The final image formed by the eyepiece is typically virtual and inverted with respect to the original object.

For viewing distant objects, the intermediate image is formed at the focal point of the objective. In normal adjustment (when the final image is formed at infinity for relaxed viewing), this intermediate image is also located at the focal point of the eyepiece. The distance between the objective and the eyepiece in this case is $L = f_o + f_e$.

The magnifying power (m) of a telescope is defined as the ratio of the angle ($\beta$) subtended by the final image at the eye to the angle ($\alpha$) subtended by the distant object at the objective (which is approximately the same as the angle subtended at the eye):

$$m = \frac{\beta}{\alpha} = \frac{f_o}{f_e}$$

A telescope with a larger focal length objective and a smaller focal length eyepiece will provide higher magnifying power.

Key considerations for astronomical telescopes are:

1. Light Gathering Power: This determines how faint an object can be observed and depends on the area of the objective lens (proportional to the square of its diameter). Larger objectives collect more light.

2. Resolving Power: This is the ability to distinguish between two closely spaced distant objects. It depends on the diameter of the objective and the wavelength of light.

Therefore, astronomical telescopes ideally have objectives with large diameters.

While refracting telescopes use lenses, large lenses are difficult and expensive to manufacture, support (they can only be supported at the edges), and they suffer from chromatic aberration (different colours focus at slightly different points). For these reasons, modern large telescopes are often reflecting telescopes, which use concave mirrors as objectives.

Advantages of reflecting telescopes:

• No chromatic aberration (mirrors reflect all wavelengths equally).
• Easier mechanical support (mirrors can be supported from the back).
• Can be made with very large apertures.

One challenge is that the image is formed inside the telescope tube, potentially requiring the observer or a secondary mirror to be placed in the light path. The Cassegrain telescope design uses a secondary convex mirror to reflect the light through a hole in the primary concave mirror, allowing the eyepiece to be placed behind the main mirror.

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